If he responds back again, suggest that he come and visit w/her for a bit and then see how things are going before suggesting to go out to eat. If she declines, then that's on him. He doesn't realize how much things have changed for her and it takes time to rebuild those relationships.
Heather, it's not about the support payments at all. It's about him trying to reconnect w/his daughter, a daughter he walked out on. He's most likely been advised that he needs to start making an effort to visit w/his children because it wouldn't look good in the eyes of the court and he doesn't know what to do, but to push. It won't work and it takes time for your daughter to feel safe around him again.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.